Convergence in total variation distance of a third order scheme for one-dimensional diffusion processes

In this paper, we study a third weak order scheme for diffusion processes which has been introduced by Alfonsi [1]. This scheme is built using cubature methods and is well defined under an abstract commutativity condition on the coefficients of the underlying diffusion process. Moreover, it has been proved in [1] that the third weak order convergence takes place for smooth test functions. First, we provide a necessary and sufficient explicit condition for the scheme to be well defined when we consider the one-dimensional case. In a second step, we use a result from [3] and prove that, under an ellipticity condition, this convergence also takes place for the total variation distance with order 3. We also give an estimate of the density function of the diffusion process and its derivatives.